Mercatus has a new book out on copyright, edited by Jerry Brito, called *Copyright Unbalanced: From Incentive to Excess*. I am pleased to be one of an otherwise-illustrious group of contributors.

I expect that the book will create some controversy in policy circles. In this post, I want to address what is likely to be a knee-jerk response from our critics, that copyright reform will substantially decrease the incentive to produce creative works.

Content creators anticipate that their products will generate some amount of revenue each year after they are released. The expectation is generally that the creative work will generate the highest revenue in the first year, and less revenue in each subsequent year. To model this revenue stream, I’m going to assume exponential decay. Exponential decay lets us pick a half-life, $h$, and assume that $h$ years after the work was released, it will generate revenue at half the initial rate. After $2h$ years, it will generate revenue at one-fourth the rate, and so on.

In year $t$, the revenue that the content creator will receive if there is copyright is $e^{\frac{-t \ln2}{h}}$ times the initial revenue. Consequently, the total revenue that a copyright holder will receive over the life of a 95-year copyright term is

$$\sum\limits_{t=0}^{94} e^{\frac{-t \ln 2}{h}}$$

times the initial revenue.

However, content creators prefer revenue now to revenue 90 years from now. In order to calculate the present value of this revenue stream, we need to apply a discount rate $r$. The ex ante value of the revenue stream generated by the 95-year copyright term is therefore

$$\sum\limits_{t=0}^{94} \dfrac{e^{\frac{-t \ln 2}{h}}}{(1+r)^t}$$

times the initial revenue.

And of course, this calculation generalizes to different copyright terms. If we returned to a 28-year term, as Tom Bell advocates in his chapter of our book, the ex ante revenue stream would be valued at

$$\sum\limits_{t=0}^{27} \dfrac{e^{\frac{-t \ln 2}{h}}}{(1+r)^t}$$

times the initial revenue.

We’re now at a point where we can start to run some numerical calculations based on plausible values for $h$ and $r$. What is a reasonable ex ante expectation about the half-life of the revenue stream of a new creative work? I expect that for our book, the half-life will be something like 1 year or less; we will probably sell less than half as many books in the second year the book is out as in the first. But let’s not use $h=1$. Let’s estimate that $h=10$ to be extremely conservative and generous to our critics.

What about $r$? Again, how about if we are conservative and give $r$ a low value, like $r=0.02$?

Now we can run some calculations. Using the values above, the ex ante present value of a 95-year copyright is around 11.726 times the initial revenue. The ex ante present value of a 28-year copyright is around 10.761 times the initial revenue. Consequently, shortening the copyright term from 95 years to 28 years (less than 30% of the current term!) retains about 91.8 percent of the incentive effect of the current copyright term.

It is unlikely that such a small decrease in the present-value of the revenue stream would reduce the amount of content production by much. To the extent that content producers cannot or do not substitute easily into other fields, they would simply take the 8.2 percent decline in compensation per project as a decrease in wages (not the end of the world), and there would be no decline in content production. To the extent that content producers can substitute into other fields, we would get less content, but we would also get more of other stuff—the welfare effects of less content are ambiguous, since there is a knowledge problem regarding the optimal amount of content.

If you want to do the calculation with different half-lives and interest rates, be my guest. I am confident that for all plausible values of $h$ and $r$, you will find that shortening the copyright term will have at most a modest effect on the incentive to create.

How about the value of the public domain? This is a little harder to model, because we care about the ex post value of works, not just the ex ante expectation that content creators have. In practice, there turn out to be works with much longer half-lives than others. This fact complicates any back-of-the-envelope calculation. We also don’t know exactly by how much content creation would fall.

But let’s abstract from this and model the value of the public domain as the revenue stream for a given project that otherwise would have gone to copyright holders above. One difference for the public domain is that it no longer makes sense to discount the stream of value—future generations aren’t sitting around, waiting to be born so that they can watch Star Wars for the first time. Therefore, normalized to our original, first-year revenue stream, an estimate of the value of the public domain under a 95-year term is

$$\sum\limits_{t=95}^{\infty} e^{\frac{-t \ln 2}{h}}$$

Under a 28-year term, the value is

$$\sum\limits_{t=28}^{\infty} e^{\frac{-t \ln 2}{h}}$$

Plugging in the value we selected earlier for $h$, 10, the former expression yields around 0.021 and the latter about 2.144. In other words, the value of the public domain would be around 100 times higher per creative work if we shortened the term to 28 years. Again, this value is highly dependent on our selection of $h$, but the reason I am doing these calculations is so that my critics can repeat them with values they find more plausible, if they so choose.

This analysis has been highly stylized, but it is also extremely conservative. The half-life of most creative works is probably much shorter than 10 years, and when valuing an uncertain revenue stream, most artists—and even content corporations—probably discount at a rate of higher than 2 percent. The value of the public domain has been understated in this analysis, because there are many works that turn out ex post to have longer half-lives (but it is still the ex ante estimate of value that matters for investment). I have also not factored in the gains from those derivative works that are impossible under the current regime due to transaction costs, or the savings in enforcement costs from having a shorter time during which enforcement is necessary, or indeed, many of the other issues discussed in our book.

I would be interested in reading further analyses like the one above from anyone who supports the current copyright term or a longer one. How do you justify such a long term? You don’t have to use my assumptions, just make your own explicit so that people can see what they are and quarrel with them. How many fewer works do you really think would be created if we shortened the term from 95 years to 28 years? Would we really be worse off? Please show your work.